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Solve Linear Inequalities

Author: Sophia
PDF Version

what's covered
In this lesson, you will learn how to solve a linear inequality. Specifically, this lesson will cover:

Table of Contents

1. Solving Inequalities

Solving inequalities is very similar to solving equations with one exception. if we multiply or divide by a negative number, the symbol will need to flip directions. We will keep that in mind as we solve inequalities.

hint
When multiplying or dividing by a negative number, the inequality sign switches. For example, greater than becomes less than, and less than becomes greater than.

EXAMPLE

Solve the inequality short dash 2 x greater or equal than 6, graph, and write in interval notation.

short dash 2 x greater or equal than 6 Divide both sides by -2
stack space short dash 2 space with bar on top space stack space short dash 2 space with bar on top Divide by a negative - flip symbol!
x less or equal than short dash 3 Graph, starting at -3, going down with a closed circle for less than or equal to
open parentheses short dash infinity comma space short dash 3 close square brackets Interval Notation

The inequality we solve can get as complex as the linear equations we solved. We will use all the same patterns to solve these inequalities as we did for solving equations. Just remember that any time we multiply or divide by a negative the symbol switches directions (multiplying or dividing by a positive does not change the symbol!)

EXAMPLE

Solve the inequality 3 open parentheses 2 x minus 4 close parentheses plus 4 x less than 4 open parentheses 3 x minus 7 close parentheses plus 8, graph, and write in interval notation.

3 open parentheses 2 x minus 4 close parentheses plus 4 x less than 4 open parentheses 3 x minus 7 close parentheses plus 8 Distribute
6 x minus 12 plus 4 x less than 12 x minus 28 plus 8 Combine like terms on both sides
10 x minus 12 less than 12 x minus 20
stack negative 10 x space space space space space space space space space minus 10 x space space space space space with bar below 		Move x to one side by subtracting 10x from both sides
short dash 12 space less than space 2 x minus 20
stack plus 20 space space space space space space space space space plus 20 with bar below 		Add 20 to both sides
8 space less than space 2 x
stack space 2 space with bar on top space space space space space stack space 2 space with bar on top 		Divide both sides by 2
4 space less than space x We can rewrite this with the x on the other side.
x greater than 4 Graph, starting at 4, going up with an open circle for less than or equal to
open parentheses 4 comma infinity close parentheses Interval Notation

summary
When graphing inequalities, it is important to be careful when the inequality is written backwards as in the above example (4 less than x rather than x greater than 4). Often students draw their graphs the wrong way when this is the case. The inequality symbol opens to the variable, this means the variable is greater than 4. So we must shade above the 4.

Solving linear inequalities is similar to solving equations, except that we use an inequality symbol instead of an equal sign. When we're solving an inequality and you multiply or divide by a negative number, your inequality symbol is going to switch directions. Also, when we're solving a compound inequality, any operation done between the inequality symbols must be done on the other side of both inequality symbols.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

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